2.2.28
Real fluids bahut messy hote hain: viscosity, vortices, turbulence. Lekin agar hum viscosity ignore kar dein (inviscid) aur ye assume karein ki fluid elements spin nahi karte (irrotational), toh velocity field ek single scalar ka gradient ban jaata hai — jaise electric field kisi potential ka gradient hota hai. Isse poora flow Laplace's equation solve karne mein aa jaata hai, jo linear hai. Linear matlab hum simple flows ko saath add kar sakte hain (superposition) aur complex flows bana sakte hain. Yahi poora fayda hai: mushkil fluid problems Lego ban jaate hain.
Definition Inviscid & Irrotational
Inviscid : viscosity μ = 0 Irrotational : vorticity ω = ∇ × v = 0 rotate nahi karte.
KYUN irrotational matter karta hai: Vector calculus ka ek theorem kehta hai ki agar kisi field ka curl zero ho (simply-connected region mein), toh woh field kisi scalar ka gradient hota hai. Toh:
∇ × v = 0 ⟹ v = ∇ ϕ
jahan ϕ velocity potential hai. Teen velocity components ki jagah ek scalar aa jaata hai.
DO ideas ko combine karne ka TARIKA. Hamare paas hai:
Irrotational ⇒ v = ∇ ϕ
Incompressible ⇒ ∇ ⋅ v = 0
Pehle ko doosre mein substitute karo:
∇ ⋅ ( ∇ ϕ ) = 0 ⟹ ∇ 2 ϕ = 0
Ye step kyun? Gradient ka divergence hi Laplacian hota hai by definition. Toh potential flow = Laplace's equation solve karna, jo physics mein sabse zyada study ki gayi linear PDE hai.
Intuition Linearity = superposition kyunLaplace's equation linear hai: agar ϕ 1 ϕ 2 ∇ 2 ϕ = 0 ϕ 1 + ϕ 2 v = ∇ ϕ velocities bhi add hoti hain . Yahi engine hai jisse simple flows se complex flows build hote hain.
Definition Stream functionψ ψ u = ∂ y ∂ ψ , v = − ∂ x ∂ ψ ψ streamlines hoti hain (fluid unke saath flow karta hai).
KYUN define karte hain: continuity automatically satisfy ho jaati hai:
∇ ⋅ v = ∂ x ∂ u + ∂ y ∂ v = ∂ x ∂ y ∂ 2 ψ − ∂ y ∂ x ∂ 2 ψ = 0. ∇ 2 ψ = 0 ( ϕ , ψ ) conjugate harmonic functions hain: ϕ = ψ =
∂ x ∂ ϕ = ∂ y ∂ ψ , ∂ y ∂ ϕ = − ∂ x ∂ ψ
Har ek ko sabse simple ϕ ψ
Worked example Rankine half-body = Uniform flow + SourceKYUN: source fluid ko outward push karta hai; dur upstream uniform flow dominate karta hai; jahan dono balance hote hain, ek stagnation point banta hai, aur ek streamline solid-looking nose jaisi ban jaati hai.
ψ = U y + 2 π m θ = U r sin θ + 2 π m θ . Ye step kyun? Hum bas dono ψ Stagnation point: v = 0 x θ = π u = U − 2 π x m = 0 ⇒ x s = − 2 π U m Kyun: source ka leftward push wahan rightward stream ko exactly cancel karta hai.
Worked example Flow over a cylinder = Uniform flow + Doubletψ = U r sin θ − 2 π κ r s i n θ = U sin θ ( r − 2 π U r κ ) . Ye step kyun? Doublet strength choose karo taaki ψ = 0 a a 2 = 2 π U κ ψ = 0 r = a θ streamline hai, yaani ek solid cylinder!
Surface speed: u θ r = a = − 2 U sin θ 2 U θ = ± 90° θ = 0 , π
Worked example Lifting cylinder = Uniform + Doublet + Vortex → Magnus liftEk vortex Γ Kutta–Joukowski theorem lift per unit span deta hai:
L ′ = ρ U Γ. Kyun matter karta hai: yahi airfoil theory ki neenv hai — spinning balls curve karte hain, wings lift karte hain.
Common mistake Common errors ko steel-man karna"Irrotational ka matlab fluid circles mein nahi jaata." Kyun sahi lagta hai: "rotation" circular motion jaisa lagta hai. Fix: free vortex perfect circles mein move karta hai phir bhi irrotational hai (∇ × v = 0 local spin of a fluid element (vorticity) ke baare mein hai, global path shape ke baare mein nahi .
"Superposition pressures bhi add karta hai." Kyun sahi lagta hai: velocities add hoti hain, toh pressure kyun nahi? Fix: Bernoulli v nonlinear hai (p + 2 1 ρ v 2 = Velocity fields add karo, phir pressure ek baar end mein total velocity se compute karo. Pressures kabhi add mat karo.
"Potential flow drag predict karta hai." Fix: body ke paas steady flow ke liye ye zero drag predict karta hai (d'Alembert's paradox) kyunki viscosity/wake nahi hai. Lift aur pressure distribution ke liye achha hai, drag ke liye useless hai.
Recall Feynman: 12-saal ke bachche ko samjhaoSocho ek syrup jisme koi stickiness nahi hai, itna smoothly flow karta hai ki koi bhi chhota piece kabhi top ki tarah spin nahi karta. Tab poore flow ko ek "height map" se describe kiya ja sakta hai — fluid hamesha is map par "downhill" roll karta hai. Kyunki map ka rule simple hai (bas add hota hai), tum ek "seedha nadi" map, ek "spreading fountain" map, aur ek "swirl" map le ke unhe ek doosre par stack kar sakte ho aur ek ball ya wing ke around flow invent kar sakte ho. Swirl ko ball-flow par stack karo aur tumne explain kar diya kyun ek spinning ball curve karta hai!
"PUSH-VD" building blocks ke liye: P otential exist karta hai kyunki flow irrotational hai; U niform, S ource/sink, V ortex, D oublet — aur ek sneaky H : doublet+uniform = cylinder tak H alf ka raasta. Yaad rakho "Velocities add hoti hain, Pressure nahi."
Potential flow define karne wale do assumptions kya hain? Inviscid (
μ = 0 ) aur irrotational (
∇ × v = 0 ).
Irrotationality humein velocity potential define karne kyun deti hai? Zero curl ⇒ field ek gradient hai:
v = ∇ ϕ .
Potential flow ki governing equation derive karo. Incompressible
∇ ⋅ v = 0 plus
v = ∇ ϕ gives
∇ 2 ϕ = 0 (Laplace).
Yahan superposition valid kyun hai? Laplace's equation linear hai, toh solutions ka sum bhi solution hai; velocities add hoti hain.
Stream function define karo aur uski key property batao. u = ∂ ψ / ∂ y , v = − ∂ ψ / ∂ x ψ =
Strength m ϕ ψ ϕ = 2 π m ln r , ψ = 2 π m θ u r = m / ( 2 π r )
Circulation Γ u θ u θ = Γ/ ( 2 π r )
Radius a Uniform flow + doublet with a 2 = κ / ( 2 π U ) r = a
Non-spinning cylinder par surface speed kya hai? u θ = − 2 U sin θ 2 U
Uniform+source (Rankine half-body) ka stagnation point kahan hai? x s = − m / ( 2 π U )
Kutta–Joukowski lift batao. L ′ = ρ U Γ
D'Alembert's paradox kya hai? Body ke paas steady potential flow zero drag deta hai (no viscosity, no wake).
Common error: kya pressures superpose hote hain? Nahi — velocity fields add karo, phir (nonlinear) Bernoulli ek baar apply karo.
Laplace's equation — woh linear PDE jo superposition ko kaam karne deti haiBernoulli's equation — superposed velocity field ko pressure mein convert karta haiVorticity and circulation — irrotationality aur vortex define karta haiKutta–Joukowski theorem — circulation se liftCauchy–Riemann equations — ϕ ψ Magnus effect — spinning cylinder/ball liftConformal mapping — in flows ko airfoil shapes tak extend karta hai
Laplace equation nabla2 phi=0
Basic building-block flows
